Phase-Space Dynamics of Runaway Electrons in Tokamaks Xiaoyin Guan, Hong Qin, and Nathaniel J

PHYSICS OF PLASMAS 17, 092502 ͑2010͒

Phase-space dynamics of runaway electrons in tokamaks Xiaoyin Guan, Hong Qin, and Nathaniel J. Fisch Princeton Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543, USA ͑Received 9 February 2010; accepted 19 July 2010; published online 3 September 2010͒ The phase-space dynamics of runaway electrons is studied, including the influence of loop voltage, radiation damping, and collisions. A theoretical model and a numerical algorithm for the runaway dynamics in phase space are developed. Instead of standard integrators, such as the Runge–Kutta method, a variational symplectic integrator is applied to simulate the long-term dynamics of a runaway electron. The variational symplectic integrator is able to globally bound the numerical error for arbitrary number of time-steps, and thus accurately track the runaway trajectory in phase space. Simulation results show that the circulating orbits of runaway electrons drift outward toward the wall, which is consistent with experimental observations. The physics of the outward drift is analyzed. It is found that the outward drift is caused by the imbalance between the increase of mechanical angular momentum and the input of toroidal angular momentum due to the parallel acceleration. An analytical expression of the outward drift velocity is derived. The knowledge of trajectory of runaway electrons in configuration space sheds light on how the electrons hit the first wall, and thus provides clues for possible remedies. © 2010 American Institute of Physics. ͓doi:10.1063/1.3476268͔

I. INTRODUCTION trajectories of runaway electrons which shifted tens of milli- meters outward and expanded radially in a single toroidal In tokamaks, relativistic runaway electrons are often ob- pass.22 More comprehensive theoretical and numerical stud- served during and after a plasma disruption or during a fast 1–4 ies are needed to understand the physics of runaway elec- plasma shutdown. The growth-rate of avalanche runaway trons drifting away from magnetic surfaces. electron induced by knock-on process, synchrotron radiation, 5–7 In this paper, we study runaway electron orbits in the and magnetic fluctuations has been studied extensively. phase space under the influences of loop voltage, synchro- Runaway electrons can hit and damage the first wall.8 In a tron radiation, bremsstrahlung radiation, and collisions in long-pulse discharge with nonzero loop voltage, runaway system time-scale. The loop voltage is modeled as an induc- electrons can hit and damage the wall even without disrup- tive electric field. The radiation damping of synchrotron and tion or fast shutdown. Also, during start-up with non-Ohmic bremsstrahlung radiation is modeled as an effective force current-drive, runaway electrons can be produced in the di- acting on electrons. Collisional effect is included using the rection opposing the current, presenting similar issues.9 Pre- Monte-Carlo methods. To numerically calculate the phase vious research had focused on the energy limit of runaway space trajectories of runaway electrons, we adopt a varia- electrons under the influence of loop voltage, synchrotron 23–25 radiation, bremsstrahlung drag forces, and collisions.10–13 tional symplectic integrator with good global conserva- The evolution path of a runaway electron in momentum tive properties over long integration time. This is because we space due to these four physical effects has been studied in need to guarantee that the numerical error accumulated over details.14,15 The effects of stochastic magnetic field on the this very long-time scale is less than the size of the physical transport and energy limit of runaway electrons have also effects. For standard numerical integrators, coherent error ac- been studied.16–20 cumulation over many time-steps can be significantly larger On the other hand, the orbits of runaway electrons in than the collisional effects and the long-term runaway dy- configuration space, even without magnetic fields fluctua- namics. Therefore, it is crucial that we use a variational sym- tion, have not been thoroughly studied. However, a compre- plectic integrator for the guiding center dynamics to carry hensive understanding of the configuration space trajectory out the numerical simulation of runaway electrons. Through of runaway electrons is important because it tells us how the simulation studies, we discover that the circulating orbits of electrons hit the first wall, and thus provides clues for pos- runaway electrons drift outward toward the wall, which is sible remedies. In addition, the termination of runaway elec- consistent with experimental observations. The physics of tron orbits in configuration space can also modify the maxi- this outward drift is analyzed. It is found that the outward mum energy, especially for those electrons originated from drift is caused by the imbalance between the increase of me- the edge region. Based on the experimental observations on chanical angular momentum and the input of toroidal angular the Tore Supra tokamak and relevant orbit calculations,21 it momentum due to the parallel acceleration. We derive an was suggested that runaway electrons can shift far away analytical expression of the outward drift velocity, which is from magnetic surfaces within one transit period, and the able to explain the main feature of the simulation results. orbits will open to intersect the conducting wall. An early The paper is organized as follows. The theoretical model study in a small tokamak indicated that there were possible of a runaway electron and the symplectic variational integra-

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tor with good global conservative properties over long inte- eA p ͩ␮ eff,Ќ Ќ ͪ gration time are developed in Sec. II. In Sec. III, numerical 2 2 − B ͱ ʈ p 2mB ␥ ͑ ͒ results are presented and the physics of the outward drift of = 1+ 2 2 + 2 . 9 the circulating orbit of a runaway electron is analyzed, along m c mc with an analytical derivation of the outward drift velocity. The collisional effects for runaway electrons are modeled by Conclusions and future work are discussed in Sec. IV. the Monte-Carlo method, where each collision induces changes in momentum space according to2,4,11 II. THEORETICAL MODEL AND NUMERICAL ALGORITHM 4 ⌳ nee ln m pʈ 2 ⌬pʈ =− ␥͑Z +1+␥͒ ⌬t = f͑pʈ,p ͒⌬t, ␲␧2 eff 3 Ќ The relativistic guiding center Lagrangian, in the ab- 4 0 p sence of the effects of radiation and collisions, can be written ͑10͒ as26 ͑ ͒ ␥ 2 ͑ ͒ 4 ⌳ ␥ L = eA0 + eA1 + pʈb · x˙ − mc , 1 2 2nee ln m 2 2 ⌬pЌ = ͓pʈ ͑Z +1͒ − ␥pЌ͔⌬t 4␲␧2 p3 eff where e is the charge, A0 is the vector potential of the equi- 0 pʈ 2 librium magnetic field, is the momentum along magnetic = g͑pʈ,pЌ͒⌬t. ͑11͒ fields, b is unit vector along magnetic fields, ␥ ͱ 2 2 2 2 ⌳ = 1+pʈ /m c +2␮B/mc is the relativistic factor, B is the Here, ln is the Coulomb logarithm. Note that the momen- 2 magnetic fields, ␮=pЌ /2mB is the magnetic momentum, pЌ tum is dominant in parallel direction, and the collisional ef- is the momentum perpendicular to the magnetic fields, m is fect is mainly the averaged slowing-down and momentum the rest mass of the electron, and c is the speed of light. flow from the parallel direction to the perpendicular direction Here, the electric field due to the loop voltage is included as due to pitch-angle scattering. an inductive field To numerically solve for the phase-space dynamics of a runaway electron, the numerical integrator we use must have Aץ 1 ͑ ͒ good global conservative properties over long integration E1 =− . 2 t time for the following two reasons. First, the time-scale ofץ We now include the effects of radiation and collisions. For evolution of the runaway electrons due to loop voltage and the radiation, previous studies have shown10,14,15 that the ra- radiation drag is much longer than the circulating period, diation can be treated as a drag force in the opposite direc- typically many thousands times of the circulating period. tion of the velocity of the particle, which is equivalent to an Standard numerical integrators, such as the fourth-order effective inductive electric field Runge–Kutta method, cannot track the trajectory accurately for such a long time.23–25 This is because these standard in- Aץ E =− eff . ͑3͒ tegrators only guarantee that the error is small in each time- -t step. However, the errors at different time-steps often accuץ eff mulate coherently and result in a large error over a large The magnitude of E is determined by eff number of time-steps. Second, the collisional effects are usu- ͑ ͒ eEeff = FS + FB, 4 ally small in each circulating period. To accurately simulate the long-term dynamics of a runaway electron and the colli- where F represents synchrotron radiation drag force, and F S B sional effects, we need to guarantee that the numerical error represents bremsstrahlung friction force. They can be ex- accumulated over this very long-time scale is less than the pressed as10,14,15 size of the physical effects. For standard numerical integra- 2 ͱ␥2 −1 3 1 sin4 ␪ tors, coherent error accumulation23–25 over many time-steps F = r mc2ͩ ͪ ␥4ͩ + ͪ, ͑5͒ S 3 e ␥ R2 r2 can be significantly larger than the collisional effects and the 0 g long-term runaway dynamics. Therefore, it is crucial that we 4 1 use a variational symplectic integrator for the guiding center F = n ͑Z +1͒mc2␥r2ͩln 2␥ − ͪ, ͑6͒ dynamics23–25 to carry out the numerical simulation of run- B 137 e eff e 3 away electrons. ␪ / ͑ ͒ where R0 is the major radius, sin =pЌ p is the pitch angle, For simplicity, we consider a two-dimensional 2D to- / 2 / ␲␧ 2 rg =pЌ eB0 is the electron gyroradius, re =e 4 0mc is the kamak model with circular concentric flux surfaces. In this classical electron radius, ne is the plasma density, and Zeff is geometry, there are two useful coordinate systems, the cylin- the effective ionic charge. Therefore, our relativistic La- drical coordinate system ͑R,␰,z͒ and the toroidal coordinate grangian for the guiding center dynamics of a runaway elec- system ͑r,␪,¯␰=−␰͒, which are illustrated in Fig. 1. The mag- tron with loop voltage and radiation effects is netic field is chosen to be ͑ ͒ ␥ 2 ͑ ͒ L = eA0 + eA1 + eAeff,ʈ + pʈb · x˙ − mc , 7 B0r B0R0 B = e␪ + e␰, ͑12͒ where the magnetic momentum and relativistic factor are qR R 2 pЌ eA ЌpЌ ␮ = + eff, , ͑8͒ where B0, R0, and q are constants with their usual meanings. 2mB 2mB The vector potentials A0, A1, Aeff,ʈ, Aeff,Ќ, which correspond

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2 ͱ 2 2 2 B0r eEeff,ʈtR0Rq r + R0q z pʈ = ͩp␰ − e − − eE tR ͪ 2q ͱ 2 2 2 1 0 RR q r + R0q 0 ͑23͒ and substitute Eq. ͑23͒ into Eqs. ͑17͒–͑21͒. The 2D dynamics r in the ͑r,␪͒ space is then determined by the Lagrangian R0 θ ␪˙ ͑ ͒ L = prr˙ + p␪ − H 24 R which is obtained from Eq. ͑17͒ by removing the p␰␰˙ term. ͑ ͒ ξ This is the Ruth reduction. The invariant p␰ enters Eq. 24 parametrically. ξ To numerically solve for the runaway dynamics we ap- ply the variational symplectic algorithm23–25 to this 2D relativistic guiding center Lagrangian. In the simulation, we use FIG. 1. 2D tokamak geometry with circular concentric flux surfaces. the Cartesian coordinates ͑x,y͒ instead of the polar coordinates ͑r,␪͒. The first-order discretized Lagrangian corresponding to L in the time interval t=͓kh,͑k+1͒h͔ is23–25 to the magnetic field, loop voltage, radiation drag force, are x − x y − y L ͑k,k +1͒ϵp ͑k,k +1͒ k+1 k + p ͑k,k +1͒ k+1 k chosen to be d x h y h B r2 R R B B R z ͑ ͒ ͑ ͒ 0 ͩ ͪ 0 0 0 0 ͑ ͒ − H k,k +1 . 25 A0 = e␰ −ln ez + eR, 13 2Rq R0 2 2R The iteration relations for each step are ץ R 0 ͑ ͒ ͓L ͑k −1,k͒ + L ͑k,k +1͔͒ =0, ͑26͒ d d ץ A1 = E1t e␰, 14 R xk

ץ ͒ ͑ Aeff,ʈ = Eeff,ʈtb, 15 ͓L ͑k −1,k͒ + L ͑k,k +1͔͒ =0. ͑27͒ d d ץ yk ͑ ͒ Aeff,Ќ = Eeff,Ќt. 16 Collisional effects are included according to Eqs. ͑10͒ and ͑ ͒ In this geometry, the relativistic guiding center Lagrangian is 11 ⌬ ͑ ͒⌬ ͑ ͒ ͑ ͒ ␥ 2 pʈ = f pʈ,pЌ t, 28 L = eA0 + eA1 + eAeff,ʈ + pʈb · x˙ − mc

␪˙ ␰˙ ͑ ͒ ⌬ ͑ ʈ ͒⌬ ͑ ͒ = prr˙ + p␪ + p␰ − H, 17 pЌ = g p ,pЌ t. 29 Equations ͑28͒ and ͑29͒ give us the iteration relations R R B R B z ͩ ͪ 0 0 ␪ 0 0 ␪ ͑ ͒ k+1 k k+1 k k+1 k pr =−e ln sin + e cos , 18 ͑pʈ − pʈ ͒ pʈ + pʈ p + p R 2 2R ͩ Ќ Ќ ͪ ͑ ͒ 0 = f , , 30 ⌬t 2 2 R R B r R B z ͩ ͪ 0 0 ␪ 0 0 ␪ k+1 k k+1 k k+1 k p␪ =−e ln cos − e sin ͑p − p ͒ pʈ + pʈ p + p R 2 2R Ќ Ќ ͩ Ќ Ќ ͪ ͑ ͒ 0 = g , , 31 ⌬t 2 2 2 ͑pʈ + eE ʈt͒r + eff, , ͑19͒ ⌬ ͱ 2 2 2 where t is the time-step. r + R0q

2 ͑ ͒ III. PHASE-SPACE DYNAMICS OF RUNAWAY B0r pʈ + eEeff,ʈt R0q R0 p␰ = ͫe + + eE t ͬR, ͑20͒ ELECTRONS 1 2Rq ͱr2 + R2q2 R 0 Using the theoretical model and variational symplectic 2 integrator developed in Sec. II, we now study the phase- pʈ 2B eE ЌtpЌ H = mc2ͱ1+ + ͩ␮ − eff, ͪ. ͑21͒ space dynamic of runaway electrons in a typical tokamak. m2c2 mc2 2mB The major radius of the tokamak is taken to be R0 =1 m, the ␰=0, we have on-axis magnetic ﬁeld is B0 =5 T, and the safety factor isץ/Lץ Because q=2. The loop voltage is chosen to be 5 V/m. The simulation L ⌬ ͑ ͒ ͑ ͒ ϫ −8/␯ץ ͑ ͒ time-step t in Eqs. 30 and 31 is chosen to be 5 10 , p␰ = = const. 22 ␯ 4 ⌳/͑ ␲␧2 2 3͒ ␰˙ where =nee ln 4 0mec is collision frequency forץ relativistic electrons. The simulation results of the phase

From Eq. ͑20͒, we can solve for pʈ in terms of the invariant space trajectory are displayed in Fig. 2. Plotted in Figs. 2͑a͒ p␰ and 2͑b͒ are runaway electron orbits in the conﬁguration

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͑ ͒͑ ͒ FIG. 2. Color online a Runaway electron drift orbits with collisions starting from the initial energy E0 =3.35 MeV, initial momentum pʈ =5mc, pЌ =1mc, ͑ ͒ ͑ ͒ ͑ ͒ and initial position x0 =0.1 m, y0 =0 m. b The same case as a , but without collisions. c Parallel momentum vs time for the same runaway electron with and without collisions. ͑d͒ Perpendicular momentum squared vs time for the same electron.

space projected onto a poloidal plane. Figure 2͑a͒ is result 3͑a͒ is the momentum trajectories of runaway electrons gen- for an electron with collisions starting from the initial energy erated in our simulation. Different lines start from different ͑ ͒ E0 =3.35 MeV, initial momentum pʈ =5mc and pЌ =1mc, initial conditions as marked in the figure. Figure 3 b is the and initial position x0 =0.1 m and y0 =0 m. The circulating momentum trajectories of the same runaway electrons ob- orbits at different time are plotted. Figure 2͑b͒ is the same tained using the numerical method reported in Refs. 14 and case as Fig. 2͑a͒, but without collisions. Plotted in Figs. 2͑c͒ 15. Our results agree with previous results very well. There and 2͑d͒ are parallel momentum and square of perpendicular are some differences in the fine scale, which can be attrib- momentum of the same electron versus time in collisional uted to the fact that in our theoretical model, the momentum and collisionless cases. These results show clearly that the space and the configuration space are coupled, and in previ- circulating orbit of the runaway electron drifts outward to- ous studies the momentum dynamics is decoupled from the ward the wall. By comparing the results with and without configuration coordinates. The agreement displayed in Fig. 3 collisions, we observe that collisions have no significant ef- indicates that the previous assumption of momentum space fects on configuration space trajectory. But in the momentum being decoupled from the configuration space is indeed valid space, collisional effects are important. Perpendicular mo- for the specific cases under investigation. mentum increases significantly as a result of the pitch-angle The outward drift of circulating orbits of runaway elec- scattering. trons is mainly due to the parallel acceleration induced by As mentioned in Sec. I, previous studies have been fo- loop voltage and radiation damping. We now analyze the cused on the runaway dynamics in the momentum space.14,15 physics of this outward drift and develop an analytical model Here, we compare our simulation results with previous stud- for it. Let us consider a guiding center orbit in a tokamak ies on momentum dynamics of runaway electrons.14,15 Figure with the circular concentric flux. Assume that we select a

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⌬p␰ =0. ͑33͒ The parallel momentum is always much larger than the perpendicular momentum, so for simplicity we only consider the radiation drag in the toroidal direction. Because the effects of the loop voltage and radiation drag in each poloidal period is small, we further assume that they can be treated as constants within each poloidal period. With these approxima- tions, Eqs. ͑32͒ and ͑33͒ can be expressed as

⌬͑␥ 2͒ Ͷ ͑ ͒ mc = e E1 + Eeff · ds

˙ Ϸ ͑ ʈ͒Ͷ ␰ ͑ ͒ − e E1 + Eeff, R dt, 34

⌬ B0x x RR0q RR0q e + ⌬pʈ + pʈ⌬ͩ ͪ q ͱ 2 2 2 2 ͱ 2 2 2 2 x + y + R0q x + y + R0q ͑ ͒ ⌬ ͑ ͒ + e E1 + Eeff,ʈ R0 t =0. 35 The integration in Eq. ͑34͒ is the toroidal precession of the circulating guiding center.27 It can be carried out along the unperturbed circulating orbit without outward drift. The ˙ quantities ␰ and p˙ ʈ needed in the integration are obtained from the unperturbed relativistic guiding center Lagrangian L0 without loop voltage and radiation drag force ͑ ͒ ␥ 2 ␪˙ ␰˙ ͑ ͒ L = eA0 + pʈb · x˙ − mc = prr˙ + p␪ + p␰ − H, 36

R R B R B z ͩ ͪ 0 0 ␪ 0 0 ␪ ͑ ͒ pr =−e ln sin + e cos , 37 R0 2 2R

2 R R B r R B z pʈr ͩ ͪ 0 0 ␪ 0 0 ␪ p␪ =−e ln cos − e sin + , R 2R ͱ 2 2 2 0 2 r + R0q ͑38͒ FIG. 3. ͑Color online͒͑a͒ Momentum space trajectories for runaway elec-

trons starting from initial momenta ͑pʈ /mc,pЌ /mc͒=͑15,3͒,͑20,4͒, 2 ͑25,5͒,͑30,6͒,͑35,7͒,͑40,8͒. ͑b͒ Momentum space trajectories for the B0r pʈR0q p␰ = ͩe + ͪR, ͑39͒ same runaway electrons using the method reported in Refs. 14 and 15. 2Rq ͱ 2 2 2 r + R0q

2 ␮ 2 pʈ 2 B ␪ H = mc ͱ1+ + . ͑40͒ ﬁxed vertical position y=r sin . At the moment the guiding m2c2 mc2 center is crossing the height of y, we measure its horizontal position, toroidal angle, and parallel momentum to be The Euler–Lagrangian equations are ͑ ␰ ͒ ץ ץ x, ,pʈ . The guiding center comes back to the same height ⌬ d L0 L0 from the same side after one poloidal period t, and the = ͑x = r,␪,␰,pʈ͒. ͑41͒ x iץ ˙xץ dt phase space coordinates are shifted to ͑x+⌬x,␰+⌬␰,pʈ i i ⌬ ͒ ⌬ ⌬ /⌬ + pʈ . Our objective is to calculate x, and x t measures From Eqs. ͑36͒–͑41͒ we have the outward drift velocity of the circulating orbit of the runaway electron. To calculate ⌬x, we use the conservation of ␰˙ = ␰˙͑0͒ + ␰˙͑1͒ cos ␪␧ + O͑␧2͒, ͑42͒ toroidal angular momentum Eq. ͑22͒, and the energy equation ͑1͒ 2 p˙ ʈ = p˙ ʈ sin ␪␧ + O͑␧ ͒, ͑43͒

2 ˙͑0͒ ͱ 2 2 2 2 ⌬͑␥ ͒ Ͷ ͑ ͒ ͑ ͒ ␰ ʈ / / ␮ / ␧ / mc = e E1 + Eeff · ds. 32 where =p R0m 1+pʈ m c +2 B0 mc and =r R0 is the inverse aspect ratio. From Eq. ͑42͒ and ͑43͒, it is clear Equation ͑32͒ states that the increase of guiding center en- that to calculate the integration in Eq. ͑34͒ to O͑␧͒,wecan ␰˙ ␰˙͑0͒ ergy equals the work done by the loop voltage and radiation use R =R0 , and pʈ =const. Therefore the toroidal preces- drag force. From Eq. ͑22͒ we have sion is

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͑ ͒ 29 pʈ⌬t Equation 47 looks similar to the Ware pinch velocity Ͷ R␰˙dt Ϸ R ␰˙͑0͒⌬t = . ͑44͒ 0 2 pʈ 2␮B E ͱ 0 1 ͑ ͒ m 1+ 2 2 + 2 vware = . 49 m c mc B␪ Substituting Eq. ͑44͒ into Eq. ͑34͒ and keeping only the lead- However, they are not the same. The Ware-pinch is the pinch ing terms in terms of ␧, we can solve for ⌬pʈ due to the effect of the toroidal electrical fields on trapped particles. Equation ͑47͒, on the other hand, describes the drift m␮B ⌬x ⌬ ͑ ͒⌬ 0 ͑ ͒ pʈ =−e E1 + Eeff,ʈ t + . 45 of circulating orbits induced by the effective toroidal electri- pʈR0 cal fields. Previously, the effect of toroidal electrical fields on ϫ Plugging Eq. ͑45͒ into Eq. ͑35͒ to eliminate ⌬pʈ and keeping circulating particles was thought to be the usual E B 29 the leading terms in terms of ␧, we have drift, i.e., E ϫ B eB0x ͑ ͒ ⌬x − e͑E + E ʈ͒x⌬t =0. ͑46͒ vdr0 = . 50 q 1 eff, B2 In above derivation, we have assumed ␳/r to be small, where In tokamaks, it can be written as ␳ is the gyroradius. This is true even for a highly relativistic EB␪ electron. For example, for an electron with momentum pʈ v = , ͑51͒ dr0 B2 ϳpЌ ϳ100mc and total energy Eϳ70 MeV, its gyroradius 0 ina5T magnetic field is only about 3 cm, which is smaller which is smaller than the Ware pinch velocity by O͑␧2͒. ͑ ͒ than the minor radius. Equation 46 states that the outward However, our analysis leading up to Eq. ͑47͒ shows that the drift velocity of the circulating orbit of a runaway electron is drift velocity of circulating orbits due to toroidal electric ϫ ⌬x q͑E + E ʈ͒ field is actually one order larger than the E B drift. It is v = = 1 eff, ͑47͒ ͑␧͒ dr ⌬t B smaller than the Ware pinch velocity by O instead of 0 O͑␧2͒. It is not small enough to be neglected in the study of which is in the x-direction. This agrees exactly with the nu- relativistic runaway electrons dynamics in tokamaks. The merical results shown in Fig. 2. From above analysis, we outward drift of circulating orbits is smaller than the Ware note that this outward drift is caused by the imbalance be- pinch because it is impossible to accelerate a trapped elec- tween the increase of mechanical angular momentum and the tron in the toroidal direction. All of the input of toroidal input of angular momentum due to the toroidal acceleration. angular momentum needs to be balanced by the change of The conservation of canonical angular moment in the toroi- radial position. For circulating electrons, most of the angular dal direction requires the radial position of the electron to momentum input is balanced by the increasing of mechanical change to compensate for the imbalance. angular momentum, and the imbalance is O͑␧͒ smaller, The trajectories of runaway electron on poloidal plane which results in an outward drift smaller than the Ware pinch have been discussed with details in previous work.28 The velocity by O͑␧͒ in amplitude. orbits are treated as closed orbits, which are described by the The analytical result in Eq. ͑47͒ is compared with the equation28 simulation result in Figs. 4͑a͒ and 4͑c͒. Plotted in Fig. 4͑a͒ is the theoretical and simulated outward drift distance versus ͑R − R − d ͒2 + z2 = const, ͑48͒ 0 net-drift time for the case of Fig. 2͑a͒. Plotted in Fig. 4͑c͒ is the where R is the horizontal position of runaway electrons, R0 is theoretical and simulated outward drift distance versus time ͑ ͒ the magnetic axis position, dnet-drift is the net-drift, i.e., the for the case of Fig. 2 b . These comparisons show that the distance between the drift orbit and the flux surface, and z is analytical model leading to Eq. ͑47͒ captures the basic fea- the vertical position of runaway electrons. The net-drift is a ture of the outward drift of the runaway electron. The effec- constant. The typical value of the net-drift is the safety factor tive electric fields of the loop voltage and radiation drag for q times the gyroradius. However, the discussion does not the same electron are plotted in Figs. 4͑b͒ and 4͑d͒ for the include the effects of toroidal electric fields, radiation, and cases corresponding to Figs. 4͑a͒ and 4͑c͒, respectively. We collisions. We have found that, under the influence of toroi- observe that when the effective electric field of radiation dal electric fields, radiation, and collisions, the orbits of run- damping is approaching to that of the loop voltage, the drift away electrons are not closed. They cannot be described by motion slows down. This is consistent with the analytical the above equation. The orbits of runaway electron drift result of Eq. ͑47͒. Again, by comparing the results with and away from flux surfaces toward to the first wall. The net drift without collisions, we observe that collisions have no signifi- is not a constant. It increases as time before the force balance cant effects on the outward drift. This is consistent with the of toroidal electric fields, radiation, and collisions is reached. results in Fig. 2. Eventually, the net-drift can be comparable to the minus ra- Finally, we emphasize that it is easy to verify that the dius. Generally speaking, the overall trajectories of runaway outward drift of runaway electrons is indeed important for electrons on poloidal plane are determined by the physics of future tokamaks operated in steady state. For nonrelativistic radiation, collisions balancing with toroidal electric fields. particles, the gyrocenter orbit is typically close to a flux sur- However, collisions are less important than radiation since face. The fact that circulating orbits of runaway electrons runaway electrons are relativistic. drift away from flux surfaces may raise the question whether

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͑ ͒͑͒ FIG. 4. Color online a The theoretical and simulated drift motion for a runaway electron which starts from the initial energy E0 =3.35 MeV, initial ͑ ͒ momentum pʈ =5mc, pЌ =1mc, and initial position x0 =0.1 m, y0 =0 m with collisions. b The effective electric ﬁeld of the loop voltage and radiation for the same electron as in ͑a͒. ͑c͒ The theoretical and simulated drift motion for the same runaway electron without collisions. ͑d͒ The effective electric ﬁeld of the loop voltage and radiation drag for the same runaway electron as in ͑c͒.

␲ / this kind of orbit is consistent with the gyrocenter approxi- for this one poloidal period is therefore 2 R0q vʈ. Because mation. In order for the gyrocenter approximation to be cor- the drift velocity relative to the closed flux surface is vd ϳ 2 /␥ rect, we only need the gyroradius to be smaller than the pʈ meeBR, with assumption that the drift velocity is a scale-length of the magnetic field. As calculated before, even constant, the maximum displacement relative to the closed for a 70 MeV runaway electron with a large perpendicular flux surface in one poloidal period can be estimated as d1 ϳ ϳ ϳ͑ ␲ / ͒͑ 2 /␥ ͒ϳ ␲ / momentum pʈ 140mc and pЌ 40mc ina2T magnetic 2 R0q vʈ pʈ meeBR 2q pʈ eB0. If the displace- field, the gyroradius is 3 cm. Therefore, for typical relativis- ment is defined as the distance between the center of the tic runaway electrons, the gyrocenter approximation is valid. runaway orbit and the magnetic axis, then it is estimated to To estimate the net departure of the orbit, we address the be28 definition of the departure first. If the displacement is defined qpʈ ϳ ͑ ͒ as the maximum displacement of a runaway electron relative d2 . 53 to the closed flux surface that it starts from in one poloidal eB0 period, then it can be estimated as This estimate is a result of toroidal angular momentum conservation. The difference between the above two estimations 2q␲pʈ ϳ ͑ ͒ d1 . 52 is caused by how the displacement is defined. There is no eB 0 real difference in terms of physics. However, we believe that The following is a simple derivation of this estimate. In one Eq. ͑53͒ should be used when assessing whether or not the poloidal period of the circulating orbit, the total distance that runaway electron reaches the wall because Eq. ͑53͒ measures ␲ the electron travels along the field line is 2 R0q. The time the net displacement of runaway electron away from mag-

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netic axis. Now we will use Eq. ͑53͒ to estimate how large this displacement can be. Considering a tokamak machine that can maintain a steady state for 10 s with magnetic ﬁeld ϳ ϳ ϳ ϳ B 2 T, safety factor q 2, Zeff 2, and loop voltage V 0, 1 V. A runaway electron in this tokamak can easily gain an

energy above 75 MeV in 2–3 s which corresponds to pʈ ϳ ϳ 150mec. The total displacement is d2 26 cm. The minor radii of Alcator C-Mod, EAST, and DIII-D are 21, 45, and 50 cm, respectively. The toroidal magnetic fields of these three tokamaks are about 5T,2T, and 2T.30–32 The estimation of ϳ d2 26 cm is comparable to the minor radii listed above. Therefore, the drift motion of runaway electron under the influence of the toroidal electric field is indeed important. In addition, even for a large tokamak such as ITER, the ϳ displacement of d2 26 cm can result in important effects. For a tokamak operated in steady state, high power RF waves will be injected into the tokamak to maintain a pre- ferred current profile. rf waves can pump electrons from FIG. 5. ͑Color online͒ Path in the configuration space of an initially trapped lower energy region to higher energy region in which run- thermal electron being accelerated to become a runaway electron under a away electrons can be generated easily. Therefore runaway loop voltage of 5 V. electrons can be born at the location where rf wave power is absorbed, for example, at some off-center location. For these runaway electrons born at off-center location, the displacement of d ϳ26 cm is large enough for them to hit the first might reach zero velocity due to collisional effects before it 2 33 wall. reaches unbounded energy. Such electrons are particularly important to consider in the case of rf heating or current drive, where the presence of the rf waves can produce run- IV. DISCUSSION AND CONCLUSIONS away electrons by accelerating particles to regions in phase The phase-space dynamics of runaway electrons in toka- space of higher runaway probability. maks has been studied in this paper. We developed a physical In addition, we further restricted our present attention to model for runaway electrons under the influence of the loop electrons whose runaway dynamics is immediate or prompt, voltage, radiation drag and collisions. In this model, we treat rather than after spending some time as thermal electrons. the loop voltage and radiation drag as inductive electric Thus we excluded the so-called backward runaways, which fields. The collisions are modeled by the Monte-Carlo are produced during noninductive start-up but change direc- method. To numerically simulate the long-term dynamics of tion before running away. We also excluded those electrons a runaway electron, we applied a variational symplectic al- that are initially trapped, and then only after several bounce gorithm for the guiding center of electrons. The variational periods become runaways. Note that the physical process symplectic algorithm adopted has good global conservative associated with such electrons can be both complex and in- properties over long integration times and thus is more suit- teresting. For example, the simulation result plotted in Fig. 5 able for simulating the runaway dynamics, compared with shows how a trapped thermal electron with an initial energy standard integrators which often have coherent error accu- 17 keV first undergoes the Ware pinch under a loop voltage mulation over a large number of time-steps. The physics of of 5 V, and then it is untrapped at a smaller radius. It subse- this outward drift is analyzed. It is found that the outward quently becomes a runaway electron under the effect of the drift is caused by the imbalance between the increase of me- same loop voltage and the circulating orbit moves outward chanical angular momentum and the input of toroidal angular toward the wall. The energy of the electron reaches 52 MeV momentum due to the parallel acceleration. An analytical at t=0.24 s. Such electrons, and for that matter the backward expression of the outward drift velocity is derived. The runaway electrons as well, may strike the tokamak limiter at knowledge of trajectories for runaway electrons in configu- a different time and at a different location than would the ration space sheds light on how the electrons hit the first wall runaway electrons that we considered, thus leaving a unique and thus provides clues for possible remedies. signature. The exploration of the paths in the configuration Note that, in this paper, we considered only the phase- space of the electrons that are not prompt runaways is, how- space dynamics of an electron after it becomes a runaway ever, left to a future study, where it is anticipated that the electron, or, in other words, we considered only those elec- symplectic algorithms developed here will be usefully appli- trons whose initial coordinates in phase space are such that cable. the probability of it running away ͑that is to reach unbounded energy͒ is essentially unity. More generally, we ACKNOWLEDGMENTS could have considered with the same formalism those electrons whose probability of running away is not necessarily This research was supported by the U.S. Department of unity, that is, that there is a finite chance that the electron Energy under Contract No. DEAC02-76-CH03073.

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